#8760920

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4377 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10,14,15)(11,16,21,17)(12,18,23,13)(19,20,22,24)(26,30)(27,33)(32,35)(34,36);(1,2)(3,7)(4,5)(6,8)(10,15)(13,18)(16,17)(20,24)(25,26,29,30)(27,31,33,38)(28,35,37,32)(34,40,36,39);(2,5)(6,7)(9,11,14,21)(10,16,15,17)(12,22,23,19)(13,20,18,24)(25,27,29,33)(26,31,30,38)(28,36,37,34)(32,40,35,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,19)(13,20)(14,21)(15,16)(18,24)(22,23)(25,27)(26,31)(28,34)(29,33)(30,38)(32,39)(35,40)(36,37);(1,2)(3,6)(4,5)(7,8)(9,12)(10,18)(11,19)(13,15)(14,23)(16,20)(17,24)(21,22)(25,28,29,37)(26,32,30,35)(27,34,33,36)(31,39,38,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,18)(10,12,15,23)(11,20,21,24)(16,22,17,19)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,37)(36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8668116);